Dragon Age: Origins: Most of your party members' efforts to rescue you from Fort Drakon which include, among other things, Morrigan the contemptuous atheist pretending to be a pious Chantry sister, Zevran scaring the guards into letting him and Oghren in with the threat that Oghren could perform the traditional Dwarven Dance of Death ("lights his pants on fire and everything!"), and Dog pretending to be sick. The version where you break yourself out also involves this if you don't just kill everything you see; you steal a uniform, say you're a new recruit, then go out on a patrol.
Call of Duty: Modern Warfare 2. The Russian plot hinges on the idea that nobody would believe that they'd launch a large-scale suicide attack against Washington, DC.
In the Blade Runner video game, one comedian goes so far as to publicly mock the protagonist. While he is on stage with him.
In Betrayal at Krondor, most of the story involves a dark elf running around in the Kingdom, which has been warring with his nation for a long time. While he makes an effort to keep his telltale pointed ears hidden, as often as not, avoiding recognition relies on the common folk assuming that anyone with pointed ears walking around openly this deep in their territory must be a light elf and not a dark elf. This is helped by the fact that there are only subtle (read: clothing, complexion, attitude) differences between dark and light elves and most common people couldn't tell one from the other.
In City of Heroes, there is a viable strategy known as "tank stealth". Tanker class PCs in the higher levels can become unkillable if they don't stick around to take damage. While stealth porter strategies commonly require that the porter in question avoid being noticed, a tank can run a map full of dangerous enemies, shrug off their fire as he passes them, and teleport the entire team to the objective point. Due to aggro rules, the enemies will most likely not chase the tank as he has yet to lock their aggro on him. The image of an 8 foot tall block of granite running through your secret underground lair is clearly a hallucination and not a catastrophic breach of security.
After discovering you're really Darth Revan, you can rub this in the face of the universe. Then you come to Lehon, where the Rakata recognize you and are upset because last time you were there you told them you needed to get into the Temple of the Ancients so you could destroy the Star Forge. While last time you were lying and took over the Temple and the Star Forge and set out to conquer the galaxy, this time it's (possibly) a reasonably accurate description of your quest. Which you can tell them. They're not inclined to believe you.
For minor dialogue hilarity (and confusion), name your character Darth Revan to begin with.
Similar dialogue hilarity occurs in any game where you give a disguised or undercover character their real name as a cover. Final Fantasy IX comes to mind with Garnet.
When you're trying to get into the Sith academy on Korriban, one of the things you can try is to Jedi Mind Trick the headmaster's apprentice into letting you in. It fails, but it still impresses her that you'd dare to try it.
The Legend of Zelda: Majora's Mask: features a game run by Zoras with the premise of destroying a large number of pots standing in a pattern with a single attack. This can indeed be done in Zora form. Alternatively, you can take Goron form and use a Powder Barrel to blow them all up. The Zoras will take it in stride and accept it as a legitimate victory.
In Assassin's Creed, when Thieves steal from people, they openly run up to them and do the deed visibly. No one protests. Then when Ezio does it sneakily, the victim can somehow recognise him and try to fight back.
In Nethack, if you pray to your patron god while they are sufficiently angry at you, they'll hit you with a One-Hit KillBolt of Divine Retribution. If you have shock resistance or reflection, you'll survive, but not to be deterred, your god will follow it up with a wide-angle disintegration beam, which is also a One-Hit Kill. It's possible to survive that as well, to which your god can only respond, "I believe it not!" and, one imagines, back away slowly and try to forget the whole thing ever happened. If you're on the Astral Plane, this won't work, as the god summons three minions to kill you the old-fashioned way.
In one Nancy Drew game, the culprits commit a crime so bizarre and specific, that it takes the police a while to find out what to charge them with. (Nancy herself seems a little confused just explaining what it was they did.)
In Mass Effect 2, Renegade Shepard can pull off some amazing acts of intimidation by simply threatening to kill people in broad daylight. At one point, a threatened asari will screech about how there's a dozen witnesses and security cameras everywhere, and Shepard will reply, "Do I look like I care about witnesses?"
The reason you can walk around with Legion on the Citadel two years after the geth attacked and killed countless people. No one thinks you're actually bringing an active geth platform onto the station; they just assume it's a fancy "personal assistance mech." Everyone is assuming the geth would rather try to infiltrate with platforms that resemble organics. When Legion points out that "Geth do not infiltrate," the security guards blow it off, and Legion then admits that "Geth do not intentionally infiltrate."
In Dragon Age II, the Mark of the Assassin DLC has Hawke and Tallis attempt to infiltrate Chateau-Haine whilst coming up with increasingly ludicrous stories to get past the guards, involving acting the part of a sterotypical foppish noble and Chewing the Scenery. Sadly, due to the guards being jaded from years of exposure to nobility (and Orlesians), it doesn't work.
Star Wars: The Old Republic: During the Republic's Makeb missions, you have to seize a giant starship from the corrupt foreman who's been embezzling from the construction funds, setting the project behind by weeks. After soundly beating him, he tries to get out of it by offering to sell you the very starship you came to take! For a 20% discount. Right after he says that, part of it falls off and crashes to the ground. "30 percent?"
Corso Riggs: Unbelievable! One of these days you have to teach me how you do that?!
Dishonored has this as an option during the Lady Boyle mission: you can enter the party, dressed as the assassin who has murdered or ruined several key political figures (by which we mean yourself) over the last few days. You can walk right up to your target, and tell her that you need to get her to the basement, since there is an assassin after her (by which we mean yourself) who is now in the building. And she will believe you, and follow you down there, because no one could ever be so brazenly obvious about being the assassin! You can even sign the guest list under your character's real name (who is, by this point, a wanted criminal), and you'll find out later that it's believed to have been a prank.